Sum frequency generator in the microwave domain for quantum communication and computation applications

ABSTRACT

A technique relates to a circuit for a sum frequency generator. A first resonator is connected to a Josephson ring modulator (JRM), and the first resonator is configured to receive a first photon at a first frequency. A second resonator is connected to the JRM, and the second resonator is configured to have a first harmonic and no second harmonic. The second resonator is configured to receive a second photon at a second frequency, and the first resonator is configured to output an up-converted photon. The up-converted photon has an up-converted frequency that is a sum of the first frequency and the second frequency.

DOMESTIC PRIORITY

This application is a continuation of U.S. application Ser. No.15/492,787, titled “SUM FREQUENCY GENERATOR IN THE MICROWAVE DOMAIN FORQUANTUM COMMUNICATION AND COMPUTATION APPLICATIONS” filed Apr. 20, 2017,which is a continuation of U.S. application Ser. No. 15/295,251, titled“SUM FREQUENCY GENERATOR IN THE MICROWAVE DOMAIN FOR QUANTUMCOMMUNICATION AND COMPUTATION APPLICATIONS” filed Oct. 17, 2016, andwhich is now U.S. Pat. No. 9,680,452. The contents of each applicationare incorporated by reference herein in their entirety.

BACKGROUND

The present invention relates to superconducting electronic devices, andmore specifically, to a sum frequency generator in the microwave domainfor quantum communication applications.

Quantum entanglement is a physical phenomenon that occurs when pairs orgroups of particles are generated or interact in ways such that thequantum state of each particle cannot be described independently of theothers, even when the particles are separated by a large distance.Instead, a quantum state must be described for the system as a whole. Toput it another way, an entangled system is defined to be one whosequantum state cannot be factored as a product of states of its localconstituents. In other words, they are not individual particles but arean inseparable whole. In entanglement, one constituent cannot be fullydescribed without considering the other(s). Note that the state of acomposite system is always expressible as a sum, or superposition, ofproducts of states of local constituents.

SUMMARY

According to one or more embodiments, a circuit for a sum frequencygenerator is provided. The circuit includes a first resonator connectedto a Josephson ring modulator (JRM), where the first resonator isconfigured to receive a first photon at a first frequency. The circuitincludes a second resonator connected to the JRM, and the secondresonator is configured to have a first harmonic and no second harmonic.The second resonator is configured to receive a second photon at asecond frequency, and the first resonator is configured to output anup-converted photon. The up-converted photon has an up-convertedfrequency that is a sum of the first frequency and the second frequency.

According to one or more embodiments, a method of forming a circuit fora sum frequency generator is provided. The method includes providing afirst resonator connected to a Josephson ring modulator (JRM), where thefirst resonator is configured to receive a first photon at a firstfrequency. The method includes providing a second resonator connected tothe JRM, and the second resonator is configured to have a first harmonicand no second harmonic. The second resonator is configured to receive asecond photon at a second frequency, and the first resonator isconfigured to output an up-converted photon. The up-converted photon hasan up-converted frequency that is a sum of the first frequency and thesecond frequency.

According to one or more embodiments, a method for remote entanglementof a first qubit and a second qubit is provided. The method includesproviding a sum frequency generator circuit separately connected to afirst quantum system and a second quantum system. The first quantumsystem includes the first qubit and the second quantum system includesthe second qubit. The method includes remotely entangling the firstqubit and the second qubit. The remote entanglement includes causing thefirst quantum system to transmit a first output readout signal at afirst frequency to the sum frequency generator and causing the secondquantum system to transmit a second output readout signal at a secondfrequency to the sum frequency generator circuit. Also, the remoteentanglement includes outputting, by the sum frequency generator, anup-converted output readout signal having an up-converted frequency thatis a sum of the first frequency and the second frequency, therebyremotely entangling the first qubit and the second qubit.

According to one or more embodiments, a method for configuring amicrowave repeater is provided. The method includes providing a firstsum frequency generator through a last sum frequency generator andproviding a first spontaneous parametric down-conversion device througha last spontaneous parametric down-conversion device. Each of the firstthrough last sum frequency generators is connected to two of the firstthrough last spontaneous parametric down-conversion devices, such thateach one of the first through last sum frequency generators is shared bytwo of the first through last spontaneous parametric down-conversiondevices. A total of the first through last sum frequency generators isone less than a total of the first through last spontaneous parametricdown-conversion devices.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a conceptual view of an application for a sum frequencygenerator (SFG) in quantum communication according to one or moreembodiments.

FIG. 2 is a schematic of an SFG circuit according to one or moreembodiments.

FIG. 3 depicts an example frequency spectrum according to one or moreembodiments.

FIG. 4 depicts an example frequency spectrum according to one or moreembodiments.

FIG. 5 depicts an example implementation of the SFG circuit according toone or more embodiments.

FIG. 6 depicts a system of an application utilizing the SFG circuit forremote entanglement between distant qubits according to one or moreembodiments.

FIG. 7 depicts a system of utilizing the SFG circuit for application asquantum microwave repeater according one to one or more embodiments.

FIG. 8 is a flow chart of a method of forming a circuit for the SFGaccording to one or more embodiments.

FIG. 9 is a flow chart of a method for remote entanglement of a firstqubit and a second qubit according to one or more embodiments.

FIG. 10 is a flow chart of a method for configuring a microwave repeateraccording to one or more embodiments.

DETAILED DESCRIPTION

Various embodiments are described herein with reference to the relateddrawings. Alternative embodiments can be devised without departing fromthe scope of this document. It is noted that various connections andpositional relationships (e.g., over, below, adjacent, etc.) are setforth between elements in the following description and in the drawings.These connections and/or positional relationships, unless specifiedotherwise, can be direct or indirect, and are not intended to belimiting in this respect. Accordingly, a coupling of entities can referto either a direct or an indirect coupling, and a positionalrelationship between entities can be a direct or indirect positionalrelationship. As an example of an indirect positional relationship,references to forming layer “A” over layer “B” include situations inwhich one or more intermediate layers (e.g., layer “C”) is between layer“A” and layer “B” as long as the relevant characteristics andfunctionalities of layer “A” and layer “B” are not substantially changedby the intermediate layer(s).

A photon is an elementary particle, which is a quantum of light alongwith all other forms of electromagnetic radiation. A photon carriesenergy proportional to the radiation frequency and has zero rest mass.

The Bell states are a concept in quantum information science andrepresent the essence of entanglement. They are subject to the Bellinequality. An EPR pair is a pair of qubits (quantum bits), particles,or photons, which are in a Bell state together, in other words,entangled with each other. Unlike classical phenomena such as theelectromagnetic and gravitational fields, entanglement is invariantunder distance of separation and is not subject to relativisticlimitations such as the speed of light. The Bell measurement is animportant concept in quantum information science. It is a jointquantum-mechanical measurement of two qubits that determines which ofthe four Bell states the two qubits are in. If the qubits were not in aBell state before, they get projected into a Bell state (according tothe projection rule of quantum measurements), and as Bell states areentangled, a Bell measurement is an entangling operation.

One useful feature of entanglement is that it can be swapped. Forexample, given two pairs of entangled photons, e.g., A, B, and C, D,where each pair is generated by a separate spontaneous photondown-converter (SPDC), it is possible to entangle the photons A and D(without having them interact with each other) by performing a jointmeasurement of photons B and C in the Bell basis and communicating theresult to A and D. One application where this quantum operation (i.e.,entanglement swapping) can be useful is quantum communication. Inparticular, it enables the implementation of quantum repeaters. However,using linear optical elements to perform partial Bell measurement inentanglement swapping schemes (in which a successful Bell measurementserves as a heralding event for the creation of entangled pair) sufferfrom several problems. For example, one problem is that spontaneousparametric down-conversion (SPDC) sources inherently emit multipairs ofentangled photons, which reduces the fidelity of the entangled stateconditioned on a successful Bell measurement and makes the entanglementswapping protocol useless without post-selecting the eventscorresponding to one detection at both ends A and D. Furthermore, inaddition to reducing the fidelity, post-selection operation by itself isincompatible with the requirements of device-independent quantum keydistribution (DIQKD) schemes (needed in order to perform secure quantumcommunications). Another problem is that all optical tests of Bell'sinequality suffer from the detection loophole where entangled photonsare not all detected due to unavoidable losses in the quantum channel,losses in the coupling between the photon-pair source and the opticalfiber, and the finite detector efficiency. Closing this loophole is arequirement for demonstrating DIQKD. One viable solution to theseproblems and others which have been proposed in the literature is usinga sum-frequency generator (which in a way serves as a nonlinear filter)instead of the linear optical elements which are used to perform theBell measurement in the entanglement swapping operation/protocol.

One or more embodiments provide a quantum device that operates in themicrowave domain (e.g., 1-30 gigahertz (GHz)). The quantum device iscapable of performing nonlinear optics operations on chip, specificallysum frequency generation, i.e., up-converting a pair of microwavephotons entering the ports of the quantum device at frequencies f_(S),f_(I) and momenta k_(S), k_(I) to an outgoing photon whose energy isequal to the sum of the energy f_(UPC)=f_(I)+f_(S) and whose momentum isequal to the sum of the momentum of the input photonsk_(UPC)=k_(I)+k_(S). One or more embodiments include a sum frequencygenerator (circuit) in the microwave domain that operates at the singlephoton level and can be utilized in various roles in quantum informationprocessing applications, particularly, in quantum computation andquantum communication.

Now turning to the figures, FIG. 1 is a conceptual view of anapplication for the sum frequency generator in quantum communicationaccording to one or more embodiments. FIG. 1 depicts SPDC 1 and SPDC 2which are two independent photon pair sources with uncorrected spectra.An SFG is connected to each SPDC 1 and 2. Single microwave photondetectors 11, 12, and 13 are respectively connected to SPDC 1, SPDC 2,and SFG.

During operation, separate pump signals are input to SPDC 1 and 2. Inthis example, pump signal 1 (at frequency ω_(P1)=ω₂+ω₄) is input to SPDC1 and pump signal 2 (at frequency ω_(P2)=ω₁+ω₃) is input to SPDC 2. TheSPDC 1 is configured to create a pair of entangled photons, for example,according to Fock states. Similarly, the SPDC 2 is configured to createa pair of entangled photons.

The entangled photon pair generated by SPDC 1 is designated as photon |1

ω₄ which is transmitted to photon detector 11 and photon |1

ω₂ which is transmitted to the SFG. The entangled photon pair generatedby SPDC 2 is designated as photon |1

ω₃ which is transmitted to photon detector 12 and photon |1

ω₁ which is transmitted to the SFG.

The SFG is configured to receive the photon |1

ω₁ and photon |1

ω₂ and up-convert the two photons (|1

ω₁ and |1

ω₂) to a converted photon |1

ω₁+ω₂. The converted photon |1

ω₁+ω₂ can be referred to as the up-converted photon. It is noted thatthe frequencies of the photons generated by SPDC 1 are ω₂ and ω₄, andthe frequencies of the photons generated by SPDC 2 are ω₁ and ω₃.Accordingly, the converted photon |1

ω₁+ω₂ is the sum frequency of ω₁+ω₂. By measuring the converted (i.e.,up-converted) photon |1

ω₁+ω₂ with single microwave photon detector, the converted photonheralds the entanglement of the of the other two photon states at adistance. In other words, measuring the converted (i.e., up-converted)photon |1

ω₁+ω₂ confirms with a certainty the entanglement of photons |1

ω₃ and |1

ω₄ which are distant from each other. Further, this scheme results in aphoton triplet state in which 3 photons are entangled. The photontriplet state of the 3 entangled photons is the entanglement of theconverted photon |1

ω₁+ω₂, the photon |1

ω₃, and the photon |1

ω₄.

FIG. 2 is a schematic of an SFG circuit 100 according to one or moreembodiments. The SFG circuit 100 is a microwave device or quantumdevice. The SFG circuit 100 includes port 150A and port 150B. The port150A can be connected to a broadband 180 degree hybrid coupler 120A, andthe port 150B can be connected to a broadband 180 degree hybrid coupler120B. The 180 degree hybrid couplers 120A and 120B each have adifference (Δ) port and a sum (Σ) port. For 180 degree hybrid coupler120A, the signal (S) is connected to the Δ port and the up-converted(UPC) signal is connected to the Σ port. For 180 degree hybrid coupler120B, the idler (I) is connected to the Δ port and a terminationimpendence point (e.g., 50 ohm (Ω) termination environment) is connectedto the Σ port.

A 180° hybrid coupler is a 4-port microwave device that is reciprocal,matched, and ideally lossless. The 180° hybrid splits an input signalinto two equal amplitude outputs. When fed from its sum port (Σ), the180° hybrid provides two equal-amplitude in-phase output signals. Whenfed from its difference port (Δ), it provides two equal amplitude 180°out-of-phase output signals.

The SFG circuit 100 includes a Josephson junction ring modulator (JRM)110. The JRM 110 includes multiple Josephson junctions (JJ) 130connected together to form the loop/ring in the JRM 110, which issimilar to a Wheatstone bridge. In one implementation, the JRM 110 canalso include JJs 131 inside the loop such that one end of each JJ 131connects to a node of the loop of the JRM 110 while the other end ofeach JJ 131 is connected to the other JJs 131. As understood by oneskilled in the art, an applied magnetic flux Φ threads the loop of theMJRM 110, and the magnetic field can be generated by a magnetic source180, such as a magnetic coil. In this example, the magnetic flux Φthreading each one of the reduced inner loops of the JRM is Φ_(ext)/4.

A signal resonator 162 includes two quarter-wavelength transmissionlines 12A and 12B. One quarter-wavelength transmission line 12A isconnected to Node A, and the other quarter-wavelength transmission line12B is connected to Node B of the JRM 110. These two quarter-wavelengthtransmission lines and the JRM 110 form a half-wavelength (λ_(S)/2)transmission line microwave resonator for the fundamental mode whosewavelength is λ_(S), which matches the wavelength of the input microwavesignal 152. The quarter-wavelength transmission lines 12A and 12B of thesignal resonator 162 connect to opposite ends of the JRM 110.

An idler resonator 161 includes two lumped-element capacitors 11A and11B each with the value the 2C_(B), where C represents capacitance. Onelumped-element capacitor 11A is coupled to node C and the otherlumped-element capacitor 11B is coupled to node D of the JRM 110. Thelumped-element capacitors 11A and 11B of the idler resonator 161 connectto opposite ends of the JRM 110.

The idler resonator 161 and the signal resonator 162 both share orutilize the JRM 110. In one implementation, both the idler resonator 161and the signal resonator 162 have the same resonance frequency asdepicted in the frequency spectrum of FIG. 3. In another implementation,the idler resonator 161 and the signal resonator 162 have differentresonance frequencies in which the idler resonator 161 has a higherresonance frequency than the signal resonator 162 as depicted in thefrequency spectrum in FIG. 4

A microwave component/element is described as lumped (versusdistributed) if its dimensions are very small compared to the wavelengthof the minimum working frequency (e.g., smaller than 1/10 of thewavelength corresponding to the minimum operational frequency of thedevice). For example, Josephson junctions are considered to a very goodapproximation, as lumped nonlinear inductors for microwave signals inthe range 1-20 GHz.

The SFG circuit 100 includes coupling capacitors 20A and 20B whichconnect the port 150A to the signal resonator 162. Also, the SFG circuit100 includes coupling capacitors 20C and 20D which connect the port 150Bto the idler resonator 161. The coupling capacitors pair 20A, 20B (andthe pair 20C, and 20D) each have the same value, and this value isdesignated Cc_(A) in coupling capacitors 20A, 20B and Cc_(B) in couplingcapacitors 20C, 20D. The value of coupling capacitors 20A, 20B, 20C, and20D is mainly determined such that it sets a desirable bandwidth for theidler resonator 161 and the signal resonator 162 (without sacrificingthe device stability as would be understood by one skilled in the art).

The idler microwave signal/tone 151 is at the frequency f_(I), and thesignal microwave signal/tone 152 is at the frequency f_(S). Theup-converted microwave signal/tone 153 is at the frequency f_(UPC). Theidler microwave signal 151 (idler photon) and the signal microwavesignal 152 (signal photon) are input into the SFG circuit 100 andup-converted to generate up-converted microwave signal 153 (up-convertedsignal).

Characteristics of the SFG circuit 100 are discussed below. The signalresonator 162 can also be referred to as resonator a, and the signalresonator 162 has the resonance frequency f_(a). The idler resonator 161can also be referred to as resonator b, and the idler resonator 161 hasthe resonance frequency f_(b). The signal resonator 162 and the idlerresonator 161 are designed to have their resonance frequencies (f_(a)and f_(b)) equal or about equal such that f_(a)˜f_(b). The idlerresonator 161 is a lumped-element resonator and does not have a secondharmonic. The signal resonator 162 is not a lumped-element resonator anddoes have a second harmonic at a resonance frequency f_(c). The secondharmonic of resonator a at the resonance frequency f_(c) satisfies therelation f_(c)=2f_(a). The second harmonic of resonator a at theresonance frequency f_(c) is designated for simplicity as the resonancefrequency of resonator c.

The frequency f_(S) (signal microwave signal 152) and the frequency f(idler microwave signal 151) are within the device bandwidth of the SFG100 and have the characteristic where they are equal or about equal suchthat f_(S)=f_(I). As an example, the frequencies can be f_(S)=f_(I)=7GHz. The frequency f_(UPC) of the up-converted signal 153 (i.e., theup-converted photon) is the sum of the frequencies f_(S) and f_(I), suchthat output up-converted frequency f_(UPC) satisfies the relationf_(UPC)=f_(S)+f_(I) (e.g., 14 GHz). The sum frequency f_(UPC) of theup-converted photon falls within the bandwidth of the 2f_(a) resonancemode.

The signal resonator 162 (resonator a) has a bandwidth designated asγ_(a), and the idler resonator 161 (resonator b) has a bandwidthdesignated as γ_(b). The second harmonic of the signal resonator 162 hasa bandwidth designated as γ_(c). In other words, resonator c hasbandwidth designated as γ_(c). The bandwidths satisfy the relationγ_(a)˜γ_(b)<γ_(c).

Also, the bandwidths satisfy the relation γ_(a), γ_(b)<g₃,γ_(2ph)<γ_(c), where g₃ is the coupling constant between the three modesa, b, and c, and characterizes the rate at which a pair of signal andidler photons are up-converted in the SFG circuit 100, where γ_(2ph) isthe rate at which signal and idler photons (via signal and idlermicrowave signals 152 and 151) leave their respective ports 150A and150B (i.e., output in reflection), and where γ_(2ph)=4 g₃ ²/γ_(c). Inthe SFG circuit 100, bandwidth γ_(c) is to be larger than the bandwidthγ_(a) and bandwidth γ_(b) such that the up-converted photon (i.e.,up-converted photon in the up-converted signal 153) leaves the SFGcircuit 100 in time, and having the larger bandwidth γ_(c) prevents theup-converted photon from having an opportunity to be down-converted. Itis noted that a 3-wave mixing process (non-linear mixing) taking placein the SFG generates the up-converted photon out of the signal photonand the idler photon.

The SFG 100 including the capacitors 11A-B and 20A-D (with the exceptionof the dielectric material in the capacitors), transmission lines 30,Josephson junctions 130, 131 (with the exception of the thin insulatingmaterial), and ports 150A and 150B are made of superconducting material.Additionally, the hybrid couplers 120A and 120B are made of low lossnormal metals or can be made of superconducting material. Examples ofsuperconducting materials (at low temperatures, such as about 10-100millikelvin (mK), or about 4 K) include niobium, aluminum, tantalum,etc.

FIG. 3 depicts an example frequency spectrum 300 according to one ormore embodiments. In this example, the resonance frequency f_(a) ofsignal resonator 162 (i.e., resonator a) and the resonance frequencyf_(b) of idler resonator 161 (resonator b) coincide with one another(i.e., are equal or about equal) as shown by the curve 305. In thefrequency spectrum 300, the bandwidths (γ_(a)˜γ_(b)) of the signalresonator 162 and the idler resonator 161 are equal or about equal.

The signal photon (which can be utilized interchangeably with the signalmicrowave signal 152) is input at the frequency f_(S), and the frequencyf_(S) is within the bandwidth γ_(a) of the signal resonator 162(resonator a). The idler photon (which can be utilized interchangeablywith the idler microwave signal 151) is input at the frequency f_(I),and the frequency f_(I) is within the bandwidth γ_(b) of the idlerresonator 161 (resonator b). in one implementation, frequency f_(S) canbe about equal to the resonance frequency f_(a), and frequency f_(I) canbe about equal to the resonance frequency f_(b).

Because the signal resonator 162 has a second harmonic resonance mode(designated as a resonance mode of resonator c) and because of theinteraction in the JRM 110, the signal and idler protons combine and areup-converted to a photon (identified as the up-converted photon of theup-converted signal 153) whose energy is the sum of the energies of thesignal and idler photons. The up-converted photon has a frequencyf_(UPC)=f_(c)=2·f_(a). In other words, the frequency f_(UPC) is at thesecond harmonic of the signal resonator 162 (resonator a), which isabout 2 times the signal resonance frequency f_(a) of the signalresonator 162. The bandwidth γ_(c) of the second harmonic (i.e.,resonator c) is about 8 times the bandwidth γ_(a) of the signalresonator 162 (resonator a), such that γ_(c)˜8·γ_(a). In anotherimplementation, the bandwidth γ_(c) can be about 7, 8, 9, and/10 timeshigher than bandwidth γ_(a). The frequency spectrum in FIG. 3 satisfiesthe relation γ_(a), γ_(b)<g₃, γ_(2ph)<γ_(c).

FIG. 4 depicts an example frequency spectrum 400 according to one ormore embodiments. In this example, the resonance frequency f_(a) ofsignal resonator 162 (i.e., resonator a) and the resonance frequencyf_(b) of idler resonator 161 (resonator b) do not coincide with oneanother as shown by the curves 405 and 410. In the frequency spectrum400, the bandwidths (γ_(a)˜γ_(b)) of the signal resonator 162 and theidler resonator 161 are separate and do not coincide with one another.

The signal photon (of the signal microwave signal 152) is input at thefrequency f_(S), and the frequency f_(S) is within the bandwidth γ_(a)of the signal resonator 162 (resonator a). The idler photon (of theidler microwave signal 151) is input at the frequency f_(I), and thefrequency f_(I) is within the bandwidth γ_(b) of the idler resonator 161(resonator b).

Because the signal resonator 162 has a second harmonic resonance mode(designated as a resonance mode of resonator c) and because of theinteraction in the JRM 110, the signal and idler protons combine and areup-converted to a photon (identified as the up-converted photon of theup-converted signal 153) whose energy is the sum of the energies of thesignal and idler photons. Unlike FIG. 3, the frequency spectrum 400 inFIG. 4 shows that the up-converted photon has a frequencyf_(UPC)=f_(c)=f_(a)+f_(b). Although f_(c) should be 2f_(a) for a uniformtransmission line resonator, the inductors (i.e., JJs 130) can skew thesecond harmonic resonance mode frequency of the signal resonator 162. Insuch a case, an embodiment can be engineered in which the resonancefrequencies f_(a) and f_(b) are different for resonators a and b andwhere their sum is f_(c). The signal resonator 162 (resonator a) can beset, for example, according to a predetermined length of thetransmission line forming the resonator 162 (and the inductive loadingof the JRM 110) to give a certain fundamental resonance frequency f_(a)and a second harmonic resonance frequency f_(c). Subsequently, the valueof the resonance frequency f_(b) of idler resonator 161 can be designedsuch that it is equal to the difference between the resonance frequencyf_(c) and f_(a).

The idler resonator 161 is structured so that its resonance frequencyf_(b) is slightly higher than the resonance frequency f_(a) of thesignal resonator 162 in order to reach the condition f_(c)=f_(a)+f_(b).For example, the resonance frequency f_(a) can be 7 GHz and theresonance frequency of the second harmonic f_(c) can be 15 GHz. In thisscenario, the resonance frequency f_(c) of the second harmonic is higherthan 2·f_(a), and in this case, f_(b)=8 GHz is engineered to be higherthan f_(a) as depicted in the frequency spectrum 400.

As discussed in FIG. 3, the bandwidth γ_(c) of the second harmonic(i.e., resonator c) is about 8 times the bandwidth γ_(a) of the signalresonator 162 (resonator a), such that γ_(c)˜8·γ_(a). In anotherimplementation, the bandwidth γ_(c) can be about 7, 8, 9, and/10 timeslarger than bandwidth γ_(a). The frequency spectrum in FIG. 4 satisfiesthe relation γ_(a), γ_(b)<g₃, γ_(2ph)<γ_(c).

For explanation purposes, a design example with feasible experimentalparameters is provided for the SFG circuit 100. The parameters includeI₀=2·10⁻⁷ amperes (A) where I₀ is the critical current of the outer JJs130 of the JRM 110 (which are nominally identical), L_(J0)=1.6 nanohenry(nH), where L_(J0) is the inductance of the outer JJs 130 for zeroapplied flux in the JRM, and L_(J)=2.3 nH where L_(J) is the inductancefor each JJ 130, for a certain working point of the device correspondingto applied flux Φ_(ext)˜Φ₀/2, where Φ₀ is the flux quantum. The innerJJs 131 are included in order to add frequency tunability to the deviceas recognized by one skilled in the art. In general, the criticalcurrent of these JJs 131 is designed to be about 2.5 times larger thanI₀ of the outer JJs 130. Additional parameters include γ_(a,b)/2π=20megahertz (MHz), γ_(c)/2π=160 MHz, f_(a)=6 GHz, f_(b)=7.3 GHz,f_(c)=13.3 GHz, C_(B)=171 femtofarads (fF), g₃/2π/=65 MHz, andκ_(2ph)/2π=105 MHz. These parameters satisfy the inequality requirementγ_(a), γ_(b)<g₃, γ_(2ph)<γ_(c).

FIG. 5 is an example implementation of the SFG circuit 100 according toone or more embodiments. In FIG. 5, the half-wave transmission lineresonator (i.e., 12A and 12B together with the JRM 110) of the signalresonator 162 is implemented as microstrips, striplines, coplanarwaveguides, etc.

The lumped-element capacitors 11A and 11B of the idler resonator 161 arecapacitors that have a common bottom plate or have common connection toa bottom plate which is not shown, and the common bottom plate is onanother level (i.e., is not coplanar with the top plates of 11A and11B). For example, dielectric material is under each top plate of thelumped-element capacitors 11A and 11B, and the lumped-element capacitors11A and 11B share a common bottom plate connected to ground. Thelumped-element capacitors 11A and 11B are connected to the JRM 110.

The half-wave transmission line resonator (i.e., 12A, 12B, and the JRM110) of signal resonator 162 is coupled to the signal and up-convertedsignal feedlines via coupling capacitors 20A and 20B, which are shown inFIG. 5 in the form of gap capacitors (other forms of capacitances arepossible, such as plate capacitance and interdigitated capacitance).Similarly, the lumped-element capacitors 11A and 11B of the idlerresonator 161 are coupled to the idler feedlines via coupling capacitors20C and 20D. The signal/up-converted signal and idler feedlines act asthe respective ports 150A and 150B which connect to the respective 180hybrid couplers 120A and 120B. The signal/up-converted signal and idlerfeedlines can be transmission lines.

FIG. 6 depicts a system 600 of an application utilizing the SFG circuit100 for remote entanglement between distant qubits 611 and 612 accordingto one or more embodiments. The system 600 includes the SFG circuit 100coupled to qubit-cavity system 601 and qubit-cavity system 602. Thequbit-cavity system 601 includes a cavity coupled to qubit 611. Thequbit-cavity system 602 includes a cavity coupled to qubit 612. Thequbit-cavity system 601 and 602 are a distance L away from each other.In one implementation, the distance L can be on the same chip such as 3cm. In another implementation, the distance L can be 1 m (meter) onseparate chips.

Example operation of the system 600 is now discussed. An input readoutsignal 605 at frequency ω_(r)+ω is input to the qubit-cavity system 601.The qubit-cavity system 601 outputs the output readout signal 605′ atfrequency ω_(r)+ω, and the SFG circuit 100 receives the output readoutsignal 605′ at frequency ω_(r)+ω (i.e., signal microwave signal 152).The output readout signal 605′ at frequency ω_(r)+ω can be input intothe A input of the hybrid coupler 120A of port 150A of SFG circuit 100.The output readout signal 605′ contains state information of the qubit611. One skilled in the art understands that the qubit-cavity system 601includes a cavity or readout resonator coupled to the qubit 611, suchthat the cavity or readout resonator transmits the output readout signal605′ in response to the input readout signal 605.

An input readout signal 610 at frequency ω_(r)−ω is input to thequbit-cavity system 602. The qubit-cavity system 602 outputs the outputreadout signal 610′ at frequency ω_(r)−ω, and the SFG circuit 100receives the output readout signal 610′ at frequency ω_(r)−ω (i.e.,idler microwave signal 151). The output readout signal 610′ at frequencyω_(r)−ω can be input into the A input of the hybrid coupler 120B of port150B of SFG circuit 100. The output readout signal 610′ contains stateinformation of the qubit 612. One skilled in the art understands thatthe qubit-cavity system 602 includes a cavity or readout resonatorcoupled to the qubit 612, such that the cavity or readout resonatortransmits the output readout signal 610′ in response to the inputreadout signal 610.

The output readout signal 605′ is interchangeable with output readoutphotons 605′, and the output readout signal 610′ is interchangeable withoutput readout photons 610′. The output readout photons 605′ can show,for example, a superposition of the energized state |e₁

and ground state |g₁

of the qubit, thereby containing the qubit state information of qubit611. The output readout photon 610′ can show, for example, asuperposition of the energized state |e₂

and ground state |g₂

of the qubit, thereby containing the qubit state information of qubit612.

In response to receiving the output readout photon 605′ at frequencyω_(r)+ω and the output readout photon 610′ at frequency ω_(r)−ω, the SFGcircuit 100 is configured to up-convert the photons 605′ and 610′,resulting in the (up-converted) converted photon 615 (output readoutsignal) at frequency 2·ω_(r). The up-converted photons 615 frequency isa sum of the frequencies (ω_(r)+ω)+(ω_(r)−ω), resulting in 2ω_(r).

The converted photons 610 (output readout signal) is a superposition ofthe following states: |e₁e₂

, |e₁g₂

, |g₁g₂

, |g₁e₂

. The converted photons 615 (or measurement of the converted photon 615)herald the remote entanglement distant qubits 611 and 612, which aredistant from each other. The converted photons 610 (i.e., theup-converted photons 153) can be output via the Σ output of the hybridcoupler 120A of port 150A of SFG circuit 100.

FIG. 7 is a system 700 of utilizing the SFG circuit 100 for applicationas a quantum microwave repeater according to one or more embodiments. Aquantum microwave repeater (or quantum repeater) is an indispensabletechnology for constructing a long-distance secure photonic network. Todistribute entanglement between two remote receivers, entanglementswapping operations at quantum repeater nodes in between are required.Accordingly, the system 700 can serve as quantum repeater nodes atpredefined locations in the communication system.

The example system 700 includes SPDC 1, SPDC 2, and SPDC 3. The SPDC 1,2, 3 can be a distance L from each other. In one implementation, theSPDC 1, 2, 3 can be a non-degenerate parametric amplifier, such as aJosephson parametric converter (JPC). Each SPDC 1, 2, 3 is coupled to anSFG circuit 100, designated as SFG 100_1 and 100_2 for explanationpurposes. Each SPDC 1, 2, 3 is an independent photon pair source withuncorrelated spectra. Each SPDC 1, 2, 3 receives its own pump signal(not shown) and then generates a pair of entangled photons.

In FIG. 700, the SPDC 1 is configured to generate entangled photons 701and 702. Photon 701 is at frequency ω₁ while photon 702 is at frequencyω₂. Photon 702 is transmitted from SPDC 1 to SFG 100_1.

The SPDC 2 is configured to generate entangled photons 703 and 704.Photon 703 is at frequency ω₃ while photon 704 is at frequency ω₄.Photon 703 is transmitted from SPDC 2 to SFG 100_1. Photon 704 istransmitted from SPDC 2 to SFG 100_2.

The SPDC 3 is configured to generate entangled photons 705 and 706.Photon 705 is at frequency ω₅ while photon 706 is at frequency ω₆.Photon 705 is transmitted from SPDC 3 to SFG 100_2.

In response to receiving photons 702 and 703 respectively at frequenciesω₂ and ω₃, the SFG 100_1 is configured to generate photon 723 atfrequency ω₂+ω₃. The SFG 100_1 transmits photon 723 to photon microwavedetector 11, where the photon microwave detector 11 detects the photon723. For the SFG 100_1, the photons 702 and 703 can be received as thesignal and idler photons 152, 151 respectively via ports 150A and 150B.

In response to receiving photons 704 and 705 respectively at frequenciesω₄ and ω₅, the SFG 100_2 is configured to generate photon 745 atfrequency ω₄+ω₅. The SFG 100_2 transmits photon 745 to photon microwavedetector 12, where the photon microwave detector 12 detects the photon745. For the SFG 100_2, the photons 704 and 705 can be received as thesignal and idler photons 152, 151 respectively via ports 150A and 150B.

The detection of photon 723 (|1

ω₂+ω₃) by photon detector 11 and the detection of photon 745 by photondetector 12 (|1

ω₄+ω₅) herald the remote entanglement of photons 701 (|1

ω₁) and 706 (|1

ω₆). The entangled photon pair 701 and 706 is created based onentanglement swapping.

It is noted that FIG. 7 shows one example of a quantum repeater setupthat includes an array of three SPDCs and two SFGs, but the setup can begeneralized/extended as necessary to N SPDCs with distance L betweenthem and N−1 SFGs, (with one SFG between two sequential SPDCs).

FIG. 8 is a flow chart 800 of a method of forming a circuit for a sumfrequency generator 100 according to one or more embodiments. At block805, a first resonator 162 (e.g., signal resonator) is connected to aJosephson ring modulator (JRM) 110 is provided. The first resonator 162is configured to receive a first photon 152 (e.g., signal microwavesignal) at a first frequency f_(S) which lies within the bandwidth ofthe fundamental resonance mode at f_(a).

At block 810, a second resonator 161 (e.g., idler resonator) isconnected to the JRM 110, and the second resonator 161 is configured tohave a first harmonic and no second harmonic. The second resonator 161is configured to receive a second photon 151 (e.g., idler microwavesignal) at a second frequency f_(I) which lies within the bandwidth ofthe fundamental resonance mode at f_(b), and the first resonator 162 isconfigured to output an up-converted photon 153 (e.g., up-convertedsignal). The up-converted photon 153 has an up-converted frequencyf_(UPC) that is a combination of the first frequency f_(S) and thesecond frequency f_(I).

A fundamental resonance frequency is about the same for the firstresonator (f_(a)) and the second resonator (f_(b)). The first frequency(f_(S)) of the first photon 152 and the second frequency (f_(I)) of thesecond photon 151 are about the same. Reference can be made to FIG. 3.

A fundamental resonance frequency of the second resonator (f_(b)) ishigher than the first resonator (f_(a)). The second frequency (k) of thesecond photon 151 is higher than the first frequency (f_(S)) of thefirst photon 152. Reference can be made to FIG. 4

The first resonator 162 has a second harmonic configured to output theup-converted photon 153 at the up-converted frequency (f_(UPC)). Theup-converted photon 153 is a sum of the energy from the first photon 152and the second photon 151.

The first resonator 162 is a half-wavelength transmission line resonator(i.e., 12A, 12B, and the JRM 110) and the second resonator 161 is alumped-element resonator (i.e., 11A, 11B, and the JRM 110). The firstresonator 162 is formed of two microstrip sections that intersect at aJRM 110 at the center. The second resonator 161 is formed of capacitors11A and 11B, each having a top plate connected to the JRM 110 and abottom plate connected together (e.g., via ground). The top plate andbottom plate are separated by a dielectric substrate or medium.

FIG. 9 is a flow chart 900 of a method for remote entanglement of afirst qubit 611 and a second qubit 612 according to one or moreembodiments. At block 905, a sum frequency generator circuit 100separately connected to a first quantum system 601 and a second quantumsystem 602 is provided. The first quantum system 601 includes the firstqubit 611 and the second quantum system 602 includes the second qubit612.

At block 910, the sum frequency generator circuit 100 is configured toremotely entangle the first qubit 611 and the second qubit 612. Byreceiving the input readout signal 605, the first quantum system 601 isconfigured to transmit a first output readout signal 605′ at a firstfrequency ω_(r)+ω to the sum frequency generator circuit 100, and byreceiving the input readout signal 610, the second quantum system 602 isconfigured to transmit a second output readout signal 610′ at a secondfrequency ω_(r)−ω to the sum frequency generator circuit 100 at block915.

At block 920, the sum frequency generator circuit 100 is configured tooutput an up-converted output readout signal 165 having an up-convertedfrequency 2ω_(r) that is a combination/summation of the first frequencyω_(r)+ω and the second frequency ω_(r)−ω, thereby remotely entanglingthe first qubit 611 and the second qubit 612.

The first output readout signal 605′ includes state information |e₁

, |g₁

of the first qubit 611 and the second output readout signal 610′includes state information |e₂

, |g₂

of the second qubit 612.

The up-converted output readout signal 615 is a superposition of thestate information |e₁e₂

, |e₁g₂

, |g₁g₂

, |g₁e₂

of the first and the second qubits 611, 612.

FIG. 10 is a flow chart 1000 of a method for configuring a microwaverepeater 700 according to one or more embodiments. At block 1005, afirst sum frequency generator through a last sum frequency generator(e.g., SFG 100_1 and 100_2) are provided. At block 1010, a firstspontaneous parametric down-conversion device through a last spontaneousparametric down-conversion device (e.g., SPDC 1, 2, and 3).

At block 1015, each of the first through last sum frequency generators(e.g., SFG 100_1 and 100_2) is connected to (i.e., receives photonsfrom) two of the first through last spontaneous parametricdown-conversion devices (e.g., SPDC 1, 2, and 3), such that each one ofthe first through last sum frequency generators is shared by two of thefirst through last spontaneous parametric down-conversion devices.

At block 1020, a total (e.g., N−1) of the first through last sumfrequency generators is one less than a total (e.g., N) of the firstthrough last spontaneous parametric down-conversion devices. It shouldbe appreciated that, although only 3 SPDC devices (i.e., N) and only 2SFG circuits (i.e., N−1) are illustrated in FIG. 7 for explanationpurposes and not limitation, N can be extended to more than 3 byanalogy.

A first photon 701 generated by the first spontaneous parametricdown-conversion device (e.g., SPDC 1) is received by none of the firstthrough last sum frequency generators (SFG 100_1 and 100_2). A lastphoton 706 generated by the last spontaneous parametric down-conversiondevice (SPDC 3) is received by none of the first through last sumfrequency generators (SFG 100_1 and 100_2). The first through last sumfrequency generators (SFG 100_1 and 100_2) are configured to causeremote entanglement of the first and second photons 701 and 706. Thefirst through last spontaneous parametric down-conversion devices (SPDC1, 2, and 3) are, for example, nondegenerate, three-wave mixingamplifiers.

Technical benefits include a quantum device that operates in themicrowave domain (e.g. 1-30 GHz). The quantum device is configured toperform sum frequency generation, i.e., up-converting a pair ofmicrowave photons entering the ports of the quantum device atfrequencies f_(S), f_(I) and momenta k_(S),k_(I) to an outgoing photonwhose energy and momentum are equal to the sum of the energyf_(UPC)=f_(I)+f_(S) and momentum of the input photonsk_(UPC)=k_(I)+k_(S). Technical benefits and advantages include remoteentanglement of two qubits where heralded entanglement generation issufficient for distributed quantum computing. The quantum device, as asum frequency generator, is a key element in device-independent quantumkey distribution schemes such as for quantum communication. The quantumdevice, as a sum frequency generator, is a key element in a quantummicrowave repeater utilized in quantum communication. Further, technicalbenefits include making the up-converted signal a resonant mode of thedevice in addition to the signal and idler by creating a hybrid type JPCthat combines microstrip resonators and lumped-element resonators.Additionally, technical benefits include designing the JRM and theelectromagnetic environment of the JRM such that the device can functionas a sum frequency generator, satisfying f_(UPC)=f_(S)+f_(I) and γ_(a),γ_(b)<g₃, γ_(2ph)<γ_(c), where γ_(2ph)=4g₃ ²/γ_(c).

The term “about” and variations thereof are intended to include thedegree of error associated with measurement of the particular quantitybased upon the equipment available at the time of filing theapplication. For example, “about” can include a range of ±8% or 5%, or2% of a given value.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which includes one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block can occur out of theorder noted in the figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments discussed herein. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdiscussed herein.

What is claimed is:
 1. A circuit comprising: a first resonator coupledto a mixing element; and a second resonator coupled to the mixingelement, the second resonator being configured to have a first harmonicand no second harmonic.
 2. The circuit of claim 1, wherein the firstresonator comprises a first frequency.
 3. The circuit of claim 2,wherein the second resonator comprises a second frequency.
 4. Thecircuit of claim 1, wherein the first resonator is configured to receivea first photon at a first frequency.
 5. The circuit of claim 4, whereinthe second resonator is configured to receive a second photon at asecond frequency.
 6. The circuit of claim 5, wherein the mixing elementis configured to receive a third frequency as a sum of the first andsecond frequencies.
 7. The circuit of claim 6, wherein the firstresonator is configured to output an up-converted photon at the thirdfrequency.
 8. The circuit of claim 5, wherein the first frequency of thefirst photon and the second frequency of the second photon are about asame.
 9. The circuit of claim 3, wherein the first frequency of thefirst resonator and the second frequency of the second resonator areabout a same.
 10. The circuit of claim 1, wherein a fundamentalresonance frequency of the second resonator is higher than the firstresonator.
 11. The circuit of claim 5, wherein the second frequency ofthe second photon is higher than the first frequency of the firstphoton.
 12. The circuit of claim 6, wherein the first resonator has asecond harmonic configured to output an up-converted photon at the thirdfrequency; and wherein the up-converted photon is a sum of energy fromthe first photon and the second photon.
 13. The circuit of claim 1,wherein the first resonator is a half-wavelength transmission lineresonator.
 14. The circuit of claim 13, wherein the second resonator isa lumped-element resonator.
 15. The circuit of claim 1, wherein thefirst resonator is formed of microstrips.
 16. The circuit of claim 1,wherein the second resonator is formed of capacitors, each having a topplate connected to the mixing element and a bottom plate connectedtogether, separated by a dielectric substrate or medium.
 17. A method offorming a circuit, the method comprising: coupling a first resonator toa mixing element; and coupling a second resonator to the mixing element,the second resonator being configured to have a first harmonic and nosecond harmonic.
 18. The method of claim 17, wherein the first resonatorcomprises a first frequency and the second resonator comprises a secondfrequency.
 19. A method for remote entanglement, the method comprising:receiving, by a circuit, a first signal associated with a first qubitand a second signal associated with a second qubit; and outputting, bythe circuit, an up-converted signal comprising first states of the firstsignal and second states of the second signal.
 20. The method of claim19, wherein the up-converted signal comprising the first states of thefirst signal and the second states of the second signal heralds a remoteentanglement of the first qubit and the second qubit.